Questions tagged [monodromy]

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A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface

Let $N(n,k)$ denote the moduli space of stable vector bundles of rank $n$ and degree $k$ over a compact Riemann surface $X$, and let $N_0(n,k)$ denote the moduli space where we fix rank $n$ and some ...
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2 votes
1 answer
134 views

The monodromy in the proof of Little Picard via Klein's $J$

First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there. ...
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  • 1,109
2 votes
0 answers
80 views

Factorization algebras as factorizable cosheaves on the (extended) Ran Space

A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...
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2 votes
1 answer
141 views

Recovering a family of rational functions from branch points

Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:X\rightarrow Y$ of degree $d$ with branch points ...
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  • 2,042
5 votes
1 answer
294 views

Thurston universe gates in knots: which invariant is it?

Today I discovered this nice video of a lecture by Thurston: https://youtu.be/daplYX6Oshc in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
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0 votes
2 answers
75 views

Tightness/Overtwistedness of genus one open book decomposition

Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})...
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  • 435
26 votes
0 answers
692 views

Nearby cycles without a function

Suppose that: $X$ is a smooth complex algebraic variety, $f : X \to D$ is a proper map to a small disc, smooth away from 0, $Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$. Then there is a procedure (...
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1 vote
1 answer
117 views

Character constructed from Kummer local system lifts to representation of algebraic torus

I'm currently reading Mar's and Springer's Character Sheaves. In Chapter 2 (Kummer local systems on tori), they provide a construction of Kummer local systems on a torus $T$ by way of the $m^{th}$ ...
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4 votes
0 answers
165 views

Criterion for triviality of monodromy in smooth families

Let $\pi: X \to \Delta^*$ be a smooth, projective morphism. We know that for each $k$, there is a natural local system $L:=R^k \pi_*\mathbb{C}$. The associated vector bundle $\mathcal{L}:=L \otimes \...
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2 votes
0 answers
153 views

Abelian variety corresponding to a vector space

I would like to know what the following statement means: "Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...
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  • 401
1 vote
0 answers
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Monodromy Representation on $H_1$ of Elliptic Curve

I'm reading this post by Charles Siegel on Monodromy Representations and there is a construction in example a not unterstand. We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic ...
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1 vote
0 answers
127 views

Moduli space of genus $g$ curves ${\mathcal{M}_g}$ irreducible by 'Monodromy argument'

I'm reading this post by Charles Siegel on Monodromy Representations and there is a short remark on the proof of irreducibility of moduli space of genus $g$ curves ${\mathcal{M}_g}$ : Just look at ${...
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2 votes
0 answers
125 views

Computing monodromy groups of curves over function fields

Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...
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1 vote
0 answers
221 views

why is monodromy weight filtration compatible with cup product?

This question is about a statement I took for granted in this question. If $f: X \to S$ is a moprhism from a complex manifold to a punctured disc then the monodromy operator $T$ is quasi-unipotent, so ...
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4 votes
0 answers
184 views

Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology

A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...
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9 votes
1 answer
497 views

A question about $p$-adic monodromy of abelian varieties

Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:...
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2 votes
0 answers
123 views

Monodromy group of the generic plane curve

Let's work over $\mathbb{C}$. The degree $d$ curves in $\mathbb{P}^2_{\mathbb{C}}$ are parameterized by a projective space $|\mathcal{O}_{\mathbb{P}^2}(d)|$. Let $U_d\subset |\mathcal{O}_{\mathbb{P}^2}...
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  • 5,613
2 votes
1 answer
173 views

Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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11 votes
3 answers
1k views

How to compute the cohomology of a local system?

Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well. Suppose that we are ...
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  • 3,716
9 votes
0 answers
219 views

Holonomy as a right adjoint, monodromy as a left adjoint

This question about the difference between holonomy and monodromy has an interesting answer by Ronnie Brown. An excerpt: So holonomy comes out as a kind of right adjoint, and monodromy as a kind ...
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  • 9,753
3 votes
1 answer
360 views

Questions about modular forms and the role of monodromy

Let $N \geq 3$ and let $\Gamma=\Gamma_1(N)$, so that the moduli problem for elliptic curves with $\Gamma$-structure is fine. Let $Y=Y(\Gamma)$ be the corresponding moduli space. In this context, one ...
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  • 643
5 votes
1 answer
662 views

Definition of geometric monodromy

Consider a polynomial $f \in \mathbb C[x_1,\dots ,x_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a topological fiber bundle. Writing $\mathrm{Atyp}(...
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  • 9,753
6 votes
2 answers
481 views

Explicit Riemann Hilbert correspondence

For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$. Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a ...
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  • 129
5 votes
1 answer
483 views

Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
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  • 1,940
1 vote
0 answers
95 views

Transverse $S^1$ actions on mapping tori

Up until now I have thought that the existence of a transverse $\mathbb{S}^1$ action on a symplectic mapping torus implies that the mapping torus is trivial. Unfortunately I also came up with a ...
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  • 939
4 votes
0 answers
117 views

Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold? The main example for this question ...
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1 vote
1 answer
305 views

The holonomy map associated to a mapping torus

So I have a rather embarrassing problem, which is not really a "problem", so much as a mental block I seem to be unable to overcome. I am trying to understand the "holonomy map" of a mapping torus. To ...
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  • 939
3 votes
0 answers
113 views

Frobenius structure for A_n singularities

I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) ...
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15 votes
1 answer
604 views

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.) Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains ...
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  • 153
5 votes
0 answers
133 views

Characterization of the hypergeometric function

One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion) In modern language (...
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4 votes
0 answers
178 views

How can I describe the monodromy of this variation of singular curves?

Consider the family of singular hyperelliptic curves $$ y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t) $$ over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes ...
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  • 1,656
3 votes
0 answers
98 views

Multivalued functions with three independent branches

Let $n$ be a positive integer and $f: \mathbb{C} \rightarrow \mathbb{C}$ be a multivalued function, analytic everywhere except for branch points at $0$, $1$ and $\infty$. Around one of those ...
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  • 263
3 votes
1 answer
152 views

How can I determine the monodromy of this variation of mixed hodge structures?

Consider the variation of mixed hodge structures which generates at the origin: $$ f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1_t $$ How can I compute ...
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  • 1,656
3 votes
0 answers
497 views

Monodromy representations are "quasi-unipotent"

Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
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4 votes
0 answers
79 views

Action of the monodromy on the cycle made of the real points

Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients. Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
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  • 1,014
1 vote
1 answer
274 views

Reference request: monodromy and isomorphic projections

I am looking for a reference to the following fact: monodromy group acting on the cohomology of smooth hyperplane sections of a smooth projective variety $X$ over $\mathbb C$ is preserved under ...
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7 votes
1 answer
309 views

Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"

In Deligne's paper on his first proof of the Weil conjectures, we have the following result. Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
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22 votes
2 answers
1k views

What are examples of D-modules that I should have in mind while learning the theory?

I've been reading about D-modules this summer in preparation for a learning seminar on intersection cohomology. Unfortunately, many of the ideas are not sticking while I learn about the theory. What ...
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  • 1,656
14 votes
2 answers
770 views

Non semi-simple monodromy in an algebraic family

I am looking for an example of a (edit: projective) family $f : X \to Y$ of complex algebraic varieties which is a topologically locally trivial fibration in (singular) varieties and such that there ...
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25 votes
1 answer
3k views

Intuition for Picard-Lefschetz formula

I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I"). To summarize the ...
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  • 2,659
7 votes
1 answer
756 views

Relations between some works by Deligne-Mostow and Thurston

Happy new year 2016! A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...
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  • 327
6 votes
4 answers
1k views

What does "higher monodromy" tell us about a principal bundle

Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to ...
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  • 7,007
8 votes
2 answers
435 views

A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...
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2 votes
0 answers
162 views

Cycle map and flat cycle

Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...
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  • 155
3 votes
0 answers
136 views

Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...
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  • 375
3 votes
2 answers
473 views

What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$. Let $H$ be the Zariski closure of the ...
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8 votes
0 answers
357 views

Degeneration of wildly ramified local monodromy representations - near or far from Deligne?

Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
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  • 116k
4 votes
0 answers
152 views

Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
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1 vote
1 answer
323 views

Monodromy of a punctured disc

I meet the following problem which I think related to the monodromy: Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, ...
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  • 3,158
1 vote
1 answer
584 views

Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...
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